The hook fusion procedure for Hecke algebras

We derive a new expression for the q-analogue of the Young symmetrizer which generate irreducible representations of the Hecke algebra. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.Comment: 19 page

The hook fusion procedure and its generalisations

The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically into ribbons. Finally, we adapt our construction to the quantum deformation of the symmetric group algebra known as the Hecke algebra.Comment: Thesis 135 page

Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element